Question

Dear Mitch,

One thing of the few things I remember my math teacher last year telling a kid who asked him for SAT advice the week before the test was "It really will help you if you have a feeling for what formulas do instead of just knowing how to use them." That teacher was a great guy but last year was his last year at the school before he retired and moved to Arizona. Anyway, the kid (and I can't even remember which kid it was) seemed to think that was really good advice. Do you happen to have any idea what it means?

Star Malone,

Allentown, PA

Answer

Dear Star,

Well I would never claim to be able to read people's minds (not even my own, sometimes!), but I think I could guess this one without even thinking very hard. Why? Because I have found myself telling students and friends either the exact same thing or something pretty close. It's easy to see within the context of a real test question, so let's look at an old favorite that still shows up a lot:

Stanley makes cookies and sells them in his bakery. It's a small store, and he only makes one type of cookie, but he makes them in two sizes. Both sizes are the same shape, circular, and the sizes are small and large. The small has a diameter of 10 centimeters, and the large has a diameter of 20 centimeters. If the small one (the 10 centimeter one) sells for 50 cents, how much should the large one with the diameter twice the size be, if Stanley is a logical businessman who did well enough in school to apply the knowledge he acquired there?

Choices:

A) $1.00

B) $1.50

C) $1.75

D) $2.00

E) $2.50

Answer?

Did you pick choice A, because it is twice the price?

Did you?

Well, it's wrong. VERY WRONG.

Why?

Because you have to ask yourself: What is the question really about? It's about food, and how much you are getting for your money. A round cookie is basically flat, so its 'amount' or size, is its surface area. And the formula for the area of a circle? Pi times the radius squared. Of course they gave you diameters, (which begins wit d for double), and diameters are double the radius. (radius is the line from center to any edge). Well, the ten centimeter one has a radius of 5, so it is pi x 5-squared, which is 25 pi.

The twenty centimeter one is pi times half of twenty (10) squared. So if the 25 pi cookie is 50 cents, the 100 pi one should be four times the price., since 100 pi is four times as much cookie as the 25 pi one. 4 x 50 cents is $2.00. That's your answer.

So to return to your initial question, I think when he teacher was talking about knowing how equations or formulas feel, you would/should get a sense that any formula that has a number in it getting squared or cubed, the size of it goes up a LOT for every drop that one of its measurements goes up. So it is not a simple linear progression. Having a feel for this type of thing will keep you alert to the answer choice that is way off.

And volume, of course even more-so because it is 3 dimensions. Make a room a few inches bigger (in other words increase each of its three dimensions a few inches, and you have MUCH LARGER unit.

And that's the kind of thing you should have a feeling for.

Hope this helps,

Mitch